Spindles have bearings in them to spin freely just like a wheel of a car. If the bearing is wearing out you will typically hear it start to whine and slowly get worse. Also, most spindles can spin freely by hand when they are turned off and you can actually feel the smoothness that they spin. Try this on a new spindle to get an idea of what's normal, spinning a bad spindle you can feel roughness or "chunkiness".
Jager spindles are pretty well known and there are many non-dental companies that can actually refurbish these spindles. I use to know a company I think in North Carolina that did a good job and was much cheaper than replacing the part with a new one. I'd start googling and calling to see what you can find.
A collet going bad is totally different than the spindle. In my experience vibrations and tool slippage (the tool will slide down as it cuts, many people will see gouged anatomy or inconsistent cutting) is a sign that a collet has gone bad or deformed. Pretty easy to replace collets if you can order the part individually. Although it should be mentioned that a tool with the incorrect shank diameter or simply a dirty collet (grease or dust) could show all these same warning signs... So this ones a bit less obvious.
Good luck!
